L(s) = 1 | + (0.142 − 0.0101i)3-s + (−0.654 − 0.755i)4-s + (−0.368 − 1.25i)5-s + (−0.969 + 0.139i)9-s + (−0.959 − 0.281i)11-s + (−0.100 − 0.100i)12-s + (−0.0653 − 0.175i)15-s + (−0.142 + 0.989i)16-s + (−0.708 + 1.10i)20-s + (−0.574 + 0.767i)23-s + (−0.601 + 0.386i)25-s + (−0.275 + 0.0600i)27-s + (−1.12 − 1.50i)31-s + (−0.139 − 0.0303i)33-s + (0.740 + 0.641i)36-s + (1.41 − 1.41i)37-s + ⋯ |
L(s) = 1 | + (0.142 − 0.0101i)3-s + (−0.654 − 0.755i)4-s + (−0.368 − 1.25i)5-s + (−0.969 + 0.139i)9-s + (−0.959 − 0.281i)11-s + (−0.100 − 0.100i)12-s + (−0.0653 − 0.175i)15-s + (−0.142 + 0.989i)16-s + (−0.708 + 1.10i)20-s + (−0.574 + 0.767i)23-s + (−0.601 + 0.386i)25-s + (−0.275 + 0.0600i)27-s + (−1.12 − 1.50i)31-s + (−0.139 − 0.0303i)33-s + (0.740 + 0.641i)36-s + (1.41 − 1.41i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4965431162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4965431162\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.959 + 0.281i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
good | 2 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 3 | \( 1 + (-0.142 + 0.0101i)T + (0.989 - 0.142i)T^{2} \) |
| 5 | \( 1 + (0.368 + 1.25i)T + (-0.841 + 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 13 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 17 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 23 | \( 1 + (0.574 - 0.767i)T + (-0.281 - 0.959i)T^{2} \) |
| 29 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 31 | \( 1 + (1.12 + 1.50i)T + (-0.281 + 0.959i)T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 41 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 43 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 47 | \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \) |
| 53 | \( 1 + (1.45 + 1.25i)T + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (1.19 + 0.0855i)T + (0.989 + 0.142i)T^{2} \) |
| 61 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 67 | \( 1 + (-1.19 + 1.37i)T + (-0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.234 + 0.797i)T + (-0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (0.755 + 0.654i)T^{2} \) |
| 97 | \( 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.476774113083334819629797635521, −9.188764137594333910421004731785, −8.170157381277718640638822820839, −7.75681189109484420347087280810, −5.99344885597816822152327846161, −5.48575171383093749403486680704, −4.69104334451561357836108718663, −3.69394101440623208029756822572, −2.11391224914818275159478963833, −0.45068107901594621129687256700,
2.65060952036104922893054858440, 3.14418331887579546734207469561, 4.23688357522998897329022960200, 5.32691809646948707714805912604, 6.44556580984062698492870388461, 7.38548613333589462323419311552, 8.030864127688147162903681312545, 8.755346459028197276288320273431, 9.737517322146844467390200562704, 10.64043260631488140979156318913